Fulleroids with dihedral symmetry

Fulleroids are cubic convex polyhedra with faces of size 5 or greater. They are suitable as models of hypothetical all-carbon molecules. In this paper, sufficient and necessary conditions for existence of fulleroids of dihedral symmetry types and with pentagonal and nn-gonal faces only depending on number nn are presented. Either infinite series of examples are found to prove existence, or nonexistence is proved using symmetry invariants.